2.7 Blending by Weight and by Volume

So far,
we have been concerned in this model only with numbers. We have not
considered units, except in a small way in the final matrix. This has
allowed us to skate over several problems where, in fact, our model has
been wrong. This does not, however, invalidate in any way what we have
learned about matrices and optimum solutions.

We have said that our qualities blend linearly. Let us consider specific gravity. Let vi be the volume of oil i. Then we know that the total volume of a blend, vTOTAL, is simply the sum of the volumes of the components, Σi vi. Also the total weight of the components, wTOTAL, equals the sum of the weights of the components, Σi wi.
But the weight of each material equals its volume multiplied by the
specific gravity multiplied by the density of water, so we have

wTOTAL = Σi wi

vTOTAL * ρ WATER * SGTOTAL = Σi vi * ρ WATER * SGi

vTOTAL * SGTOTAL = Σi vi * SGi

This last line demonstrates that specific gravity blends linearly by volume.

Now consider Sulphur content. This is expressed as a percentage by weight. So the weight of Sulphur in any component i is wi * 0.01 * SULi.

Now clearly the total quantity of Sulphur in a blend is the sum of the quantities in the components, so we have

wTOTAL * 0.01 * SULTOTAL = Σi wi * 0.01 * SULi

We can multiply throughout by 100 to get

wTOTAL * SULTOTAL = Σi wi * SULi

which is a statement that Sulphur content blends linearly by weight.

But
what did we do in our matrices? We simply said that all the qualities
blended linearly. By ignoring the units, we allowed ourselves to miss
the essential difference between blending of specific gravity and of
Sulphur content. It so happens that VBI, like Sulphur content, blends
by weight, although there are some which blend by more exotic
quantities such as mol or pseudo-volume.

In
a real model, we would correct this mistake. But here it suffices to
observe that we must be careful and precise. We must define the units
of all the elements in our model - data, decision variables,
constraints - and ensure dimensional consistency throughout. Apparently
trivial alterations to an MP model can have dramatic effects on the
optimal solutions.